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ISSN 1088-6850(e) ISSN 0002-9947(p)

Small deviations of weighted fractional processes and average non-linear approximation

Author(s): Mikhail A. Lifshits; Werner Linde
Journal: Trans. Amer. Math. Soc. 357 (2005), 2059-2079.
MSC (2000): Primary 60G15; Secondary 47B06, 47B10, 47G10
Posted: December 9, 2004
MathSciNet review: 2115091
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Abstract | References | Similar articles | Additional information

Abstract: We investigate the small deviation problem for weighted fractional Brownian motions in -norm, $1\le q\le\infty$ . Let be a fractional Brownian motion with Hurst index . If , then our main result asserts

$\begin{displaymath}\lim_{\varepsilon\to 0} \varepsilon^{1/H}\log \mathbb{P}\left... ...c(H,q)\cdot\left\Vert{\rho}\right\Vert _{L_r(0,\infty)}^{1/H}, \end{displaymath}$

provided the weight function $\rho$ satisfies a condition slightly stronger than the

-integrability. Thus we extend earlier results for Brownian motion, i.e.

, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non-linear approximation technique for Gaussian processes as well as sharp entropy estimates for

-sums of linear operators defined on a Hilbert space.

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